Non linear constitutive models for lattice materials

TLDR: In this paper, a non-linear constitutive model for lattice materials is proposed, where a representative volume element (RVE) of the lattice is modelled by means of discrete structural elements, and macroscopic stress-strain relationships are numerically evaluated after applying appropriate periodic boundary conditions to the RVE.

Abstract: We use a computational homogenisation approach to derive a non linear constitutive model for lattice materials. A representative volume element (RVE) of the lattice is modelled by means of discrete structural elements, and macroscopic stress–strain relationships are numerically evaluated after applying appropriate periodic boundary conditions to the RVE. The influence of the choice of the RVE on the predictions of the model is discussed. The model has been used for the analysis of the hexagonal and the triangulated lattices subjected to large strains. The fidelity of the model has been demonstrated by analysing a plate with a central hole under prescribed in plane compressive and tensile loads, and then comparing the results from the discrete and the homogenised models.

Figures

Figure 1: Multiscale scheme. At each integration point of the macroscopic model the constitutive relations are found by solving a boundary value problem (b.v.p) on a RVE of the lattice.

Figure 14: The deformed configurations of the triangulated lattice plate at an applied tensile strain n = 0.01 using the (a) homogenised and (b) discrete (LX/L0 = 30) models. Contours of |G| are included in (a). (c) A comparison between the predicted deformed cell shapes at 3 locations marked A, B and C. The undeformed lattice is shown in cyan as a dotted line, while the deformed lattice is in blue solid line. In each case the image on the left shows the deformed cells shapes as predicted by the discrete model while the image on the right is the RVE in the homogenised model.

Figure 6: The predicted uniaxial stressing responses of the hexagonal lattice in the (a) 1-direction and (b) 2-direction. Results are included for a range of RVE sizes while the deformed RVE shapes are shown for the 1×1 UC and 3×3 UC cases at selected values of the applied strains.

Figure 2: Sample lattice topologies. Thick blue lines represent the unit cell elements; dashed red lines describe RVE boundaries while a1 and a2 are the tessellation vectors. X1 and X2 are the coordinate directions of the Cartesian reference system.

Figure 5: (a) The predicted equi-biaixal normalised stress versus strain curve for the hexagonal lattice for selected choices of the RVE size. (b) The effect of RVE size on the predicted buckling modes three selected boundary conditions (uniaxial compressive straining in the 1 and 2 -directions, and equi-biaxial compressive straining). The undeformed lattice is shown in cyan as a dotted line, while the deformed lattice is in blue solid line.

Figure 15: The deformed configurations of the hexagonal lattice plate at an applied compressive strain n = −0.01 using the (a) homogenised and (b) discrete (LX/L0 = 30) models. Contours of |G| are included in (a). (c) A comparison between the predicted deformed cell shapes at 3 locations marked A, B and C. The undeformed lattice is shown in cyan as a dotted line, while the deformed lattice is in blue solid line. In each case the image on the left shows the deformed cells shapes as predicted by the discrete model while the image on the right is the RVE in the homogenised model.

Frequently asked questions

Q1. What are the contributions in "Non linear constitutive models for lattice materials" ?

Pasini et al. this paper used a computational homogenization approach to derive a non linear constitutive model for lattice materials. 

Q2. How is the first Piola-Kirchoff tensor evaluated?

At every integration point of the macroscopic model, the first Piola-Kirchoff tensor, P, is evaluated by means of a finite element model of the RVE. 

Q3. What is the relative density of the hexagonal and triangulated lattices?

In terms of their slenderness ratio λ, the relative density of the hexagonal and triangulated lattices are ρ∗ = 2/ √ 3λ and ρ∗ = 2 √3λ, respectively. 

Q4. What is the effect of the reorientation of the struts on the la?

The homogenised model of the hexagonal lattice could capture its typical compliant behaviour in compression, and the stiffening effect due to the reorientation of the struts along the load direction in tension. 

Q5. What is the buckling mode of the hexagonal lattice?

From these observations, the authors conclude that the microscopic buckling of the hexagonal lattice is effectively governed by the modes with a wavelength of two UCs for a wide range of loading states and a 2x2UC RVE should suffice to model the hexagonal lattice as a homogenised continuum. 

Q6. What is the buckling mode for the triangulated lattice?

The authors note that similar to the honeycomb, the triangulated lattice does not display a bifurcation under tensile loading in the 1-direction, but under compression the struts aligned with the loading direction buckle as shown in Figure 8a.